This is kind of embarrassing, but I never figured out how to cite journal names in the bibliography, especially when to abbreviate and how. For example, do we write "Adv. in Math." or "Advances in Math."? Or is "Trans. of AMS" OK? So:

Is there any formal guideline on how to cite and abbreviate journals names in a math paper?

I googled and found one resources which says: follow Math Sci Net. But it is not very convenient, since I would have to go to Math Sci Net and search for such journal every time I need to cite it.

And while we are at it, what is the protocol on arxiv links? Do we keep them in the citation after the paper is published, since it is still the most accessible source?

My impression was that this is supposed to vary according to the house style of the publisher; however, this is not always applied consistently
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Yemon ChoiMar 8 '11 at 4:50

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What's the point of abbreviating titles of journals? Does it hurt to write them in full, so that the readers don't have to struggle with bizarre cryptic abbreviations like Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. (an actual example found in the link above)? The days of paper journals are gone and we can now afford to write names in full. And if some journal still abbreviates titles, their copy-editors will always do the work for you.
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Dmitri PavlovMar 8 '11 at 6:04

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"The days of paper journals are gone" - you will have to tell the journals that, Dmitri. Good luck publishing only in morally acceptable, tree-friendly places
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Yemon ChoiMar 8 '11 at 6:31

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@Dmitri: in principle you are right, but one advantage of abbreviating journal names in a uniform manner is that that's how they are cataloged in libraries. I know you are thinking electronic journals, but I think that people are still using results that were published in the last century or even earlier and some of those journals (or at least their older years) are not yet available electronically. I've had several occasions when I was searching for a journal knowing its entire name, but I could not find it on the shelf. The librarians fortunately could help....
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Sándor KovácsMar 8 '11 at 7:07

4 Answers
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Hailong, I would still suggest following MathSciNet, but you don't necessarily have to go there every time. Presumably the journals you cite come from a small finite set. This may be a pain at first, but you will get them all very soon.

Actually, I use BibTeX and so I have a growing database of papers. You can go to MathSciNet and ask it to display references in BibTeX format. Then you can copy and paste. After a while most papers you cite are already in your database, except recent ones, but those you can get when needed. It's pretty convenient and easy. You could also use John Palmieri's bibweb script.

As for the arXiv links, I am not sure that you need to keep them there. If the author and title are the same, which is usually the case, then it is fairly easy to find on arxiv, so if someone does not have access to the published version, then they can easily find it on arXiv.org. If you want to be thorough you can include the doi information and/or the MR number.

I strongly agree with those that already suggested to use the standard abbreviations (used in MathSciNet and Zentralblatt Math.), which is what various mathematical journals explicitly suggest/requier (I believe it is not always enforced though).

Some (additional) reasons.

a. While it is save to assume that every human wanting to read your paper will understand what "Trans. of AMS" means, this is much less clear for "non-human readers" who might only know the official abbreviation "Trans. Amer. Math. Soc.".
For example in MathSciNet lists of references are (in part) included and cross-linked;
also some journals do so, and then there are providers of citation statistics (which, as long as they exist and are used, should at least be accurate). It is my understanding that all this is done automatically, to the extent possible. The usage of various different abbreviations makes this task significantly harder.

b. As said for your concrete example "Trans. of AMS" I think there is little risk of confusion.
Even, if you take this one step further and just write "TAMS", chance are people will understand. However, for slightly less universially known journals; things start to become less clear, and even then it can be confusing. It once took me a while to figure out that "CPC" refers to "Combinatorics, Probability and Computing" with the official abbreviation "Combin. Probab. Comput." I would not have had any problem.
(And, when I had that problem, I was very well aware of the existence of this journal, it was just I could not link it instantly to "CPC".)

c. Why, then, not full names? I have no strong opinion against that. But:
i. it is not what journals ask for.
ii. they can be really long.
iii. some journals are not so well-known under their full name.
Again a personal anecdote: In a list of journals (all kinds of subjects, but an incomplete list) I was searching for the journal "Combinatorica", what I found was the full name "Combinatorica. An International Journal on Combinatorics and the Theory of Computing".
Before comparing the ISSN-number I was uncertain whether this is the journal I am looking for, or something else. (Well, I could and should have known the full name, but I never noticed it before.)

d. A journal that is very well-known now, and thus will be recognized by a short ad-hoc abbreviation, can stop to exist or be renamed.
Then, perhaps, in several decades (and this is not an unreasonably long time for a mathematical paper) this journal will be not well-known anymore, and an informal abbreviation will cause trouble. (With an official one, one can always look it up.)

e. Some journals are only very slightly renamed, different series and alike. Even worse, different journals have similar names. Or, conversely, while the journal is renamed the abbreviation is kept the same (for continuity).
If one is not careful one might miss these small difference when making up ones own abbreviations.
To give an example:
'Archiv der Mathematik' is officially abbreviated 'Arch. Math. (Basel)'.
'Archivum Mathematicum (Brno)' is officially abbreviated 'Arch. Math. (Brno)'.
In case one should only know one of them 'Arch. Math.' might seem the natural abbreviation for either of them.
(The later is also an example for another thing mentioned in this point and also c.: the name I gave was the official (long) name from 88-90; before that it was 'Spisy Přírodovědecké Fakulty University J. E. Purkyně v Brně. Archivum Mathematicum' and now it is 'Universitatis Masarykianae Brunensis. Facultas Scientiarum Naturalium. Archivum Mathematicum')

Dear unknown, thank you for your answer. I agree that the standard abbreviation should be used. My problem was I didn't know where to find it.
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Hailong DaoMar 8 '11 at 16:02

Oh, sorry, I misinterpreted the question; I guess since there was already a bit of discussion going on when I arrived. Since it was not yet given, the Zentralblatt abbreviations can be found here zentralblatt-math.org/zmath/en/journals (browse and search facility). Sometimes (though rarely I believe) they differ from MathSciNet (Commun. Algebra vs Comm. Algebra is an example I know). While I typically use MathSciNet, Zentralblatt (which also provides bib-tex data) can be convenient when being without subscription to MathSciNet (it is more generous to non-subscribers).
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quidMar 8 '11 at 17:10

I'd suggest to always write all names in full. Extra clarity never hurts, and the few characters saved by abbreviations are hardly worth it, especially if the preprint is distributed in an electronic form. If you submit it to a journal, their staff will take care of imposing their house style on the references.

But the true full names (as given in MathSciNet) are sometimes really long; and, I would say, hardly anybody knows them and sometimes it would rather cause confusion. In general, I agree that this is better than using ad-hoc abbreviations (though I prefer official abbreviation). For many journals full name works well, but there are some exceptions (see c. of my answer for one example and, now also, e. for another one).
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quidMar 8 '11 at 13:45

I agree with the other answers suggesting to use the standard abbreviations. The question is of course, Which are the standard abbreviations? Just recently (while arguing with the Springer Correction Team about, well, abbreviations of journal titles) I learned that for example Springer uses the abbreviations as given by the ISSN database. To me this seems indeed somewhat more standard and moreover more universal than any other list. Unfortunaetly, this list seems not accessible to mere mortals. But said page can, with some care, at least be partially useful, as it provides this accessible list of standard abbreviations of words that may occur in titles of serials.